Numerical simulation and experiment of double chamber brake based on CFD

The artillery firing process will instantly produce high-temperature and high-pressure gunpowder gas, this process will produce shock waves. The gunpowder gas has a limited effect on the projectile during the firing and ballistic after-effects period, however, it has a very obvious effect on the stability of the gun body, and the reduction of the stability of the gun body directly affects the firing accuracy and the safety of the firing personnel. Based on the method of Computational Fluid Dynamics (CFD), numerical simulation is carried out, and the structure and flow parameters of the muzzle flow field are obtained by using three-dimensional Euler's control equation, gas equation of state, and k-epsilon model, as well as dynamic mesh technology. By comparing the flow parameters of the brake before and after optimization, and analyzing the results obtained from the 8-round firing experiments, the efficiency of the optimized brake is increased by 8.2%, and the deviation between the experimental data and the simulation results is only 10.5%, which not only verifies the accuracy of the numerical simulation calculations but also verifies the optimized brake's good retracting effect. The results of the study can provide a reference for the optimization and design of the double-chamber brake.

www.nature.com/scientificreports/optimized numerical simulation results are compared with the experimental results to verify the accurate availability of the numerical simulation simulation calculation.

Modeling
Figure 1a shows the conventional brake model.The central single hole of the baffle scheme is 22 mm in diameter, with 2 chambers and 2 rows of side apertures, and the wall of the side aperture is at an angle of 72° to the bore axis.The width of the baffle is 85 mm, the height is 60 mm, and the total length is 166 mm.
As shown in Fig. 1b, the model of the optimized brake is shown.The diameter of the central bullet hole is still 22 mm, 2 recoil chambers are retained, 2 rows of side apertures, and the angle between the side apertures and the bore axis is changed to 60°.The maximum height of the brake is 60 mm, the width is set to 90 mm, the height of the brake cavity is 52 mm, and the total length is extended to 180 mm.
The adjustment of the angle between the side aperture and the bore axis of the optimized type of brake makes the expanded powder gas more efficiently discharged to the side aperture, reduces the backward axial reaction force and further increases the forward lateral reaction force, which is a reference value for the study of the improvement of the efficiency of the brake.
A model of the computational grid area is shown in Fig. 2a,b.The gun muzzle flow field is external to the atmospheric environment and has no actual boundary.For such problems, it can be considered that a suitable external spatial region is selected as the study area.In the process of artillery firing, the high-pressure and violently burning gunpowder gas is ejected from the muzzle section and expands violently, interacting with the atmosphere to form a complex gas jet flow field.The flow field will show the complex structure of the excitation system such as reflection intersection and Mach disk, and the area of the flow field will become larger and larger  1.The flow is quasi-constant.2. The flow is one-dimensional, even if the cross-sectional area of the flow varies, the radial fractional velocity of the flow and the effect of the inhomogeneity of the flow parameters in the cross-section are corrected by the corresponding coefficients.3. The flow is isentropic, i.e., the heat exchange between the airflow and the chamber wall is neglected, and the friction caused by the viscosity of the gas is neglected.The resulting errors are also corrected by the coefficients.4. The gunpowder gas is regarded as a complete gas (or ideal gas) and the mass force is neglected.
Under the above assumptions, the three-dimensional constant isentropic flow theory in gas dynamics can be applied to study the flow-void problem of gunpowder gas in the post-effect period.

Control equations
According to the above assumptions, the Euler equation for three-dimensional non-constant compressible flow in the gun bore can be established under the premise of continuous medium mechanics.

Among them:
The equation of state and auxiliary equations are: (1) Comparison of pressure at the bore section in the case of dense grid.
where ρ is the gas density; E is the total energy of the gas; u, v, w is the velocity component of the gas along the x, y, z direction, respectively; p is the gas flow pressure; γ is the specific heat ratio; γ is the enthalpy of the gas flow; c is the local speed of sound.In this paper, the coordinate system is established with the projectile launch direction as the positive direction of the coordinate axis and the moment of projectile movement to the rifling port as the initial time, i.e., T = 0.
Gas pressure calculation mainly requires solving five unknown quantities of pressure, temperature, density,velocity of the piston and displacement, here we establish a mathematical model by gas motion and piston motion for solving.In this paper, we take time T as the independent variable and the dependent variable as the previous five unknown quantities and establish the equation for the joint solution.The expressions are as follows: 1.The equation of the variation law of gas density inside the gas conduction chamber: in the formula: 2. Pressure change equation of gas chamber:

Equation of state:
The meaning of each symbol in the formula is: γ −1 p q ρ q Reverse critical flow: www.nature.com/scientificreports/Under the assumption of continuous medium mechanics, airflow can be described by the three-dimensional Euler equation with the gas equation of state.In calculating flow under hypersonic conditions, the Reynolds number is so high that the viscosity of the gas can usually be neglected.The flow of fluids is governed by physical conservation laws, and the above control equations are mathematical descriptions of these conservation laws.Computational mechanics of fluids is based on the above control equations to accurately calculate the flow of fluids.

Initial conditions
This simulated gun uses a single charge, and the set of ballistic equations within the single charge can be summarized as follows: 1. Shape function: 2. Combustion velocity equation:

Projectile equation of motion:
Equation of motion expressed in terms of average pressure and secondary work factor ψ:

Basic equations of internal ballistics:
in the formula: In this paper, the rifling end face is set as the entrance boundary, and the initial conditions and flow parameters of the powder gas are obtained by solving the internal ballistic program.The projectile trajectory equations are given by the UDF preparation.The exit boundary is set as a pressure exit and the rest is given as a solid wall boundary.The internal ballistic time velocity curve and time rifling pressure curve are shown in Figs. 4 and 5.
Combining the results of the internal ballistic calculation above and the model of the computational grid area in Fig. 6 (where the schematic diagram can be obtained according to Fig. 7), the modal simulation initial conditions and the experimental initial conditions are plotted in a Table 1.
area between piston and gas chamber V q0 − Initial volume of the gas conductivity chamber S h − Piston cross−sectional area P p − Pressure of inflowing gas into the pilot chamber ρ p − Density of the inflow gas into the conductive chamber T p − Temperature of the inflow gas into the conductive chamber dQ dt − Pressure of the gas in the gas conductor chamber R − Gas constants ρ q − Gas density inside the gas conductivity chamber T q − Gas density inside the gas conductivity chamber F q − Air chamber surface area F q0 − Initial surface area of air chamber The sum of the masses of the moving parts with the piston and the heat loss

Numerical simulation of the breechless brake
Numerical calculations based on the computational fluid dynamics method are carried out to obtain the bore flow parameters and the bore side aperture and central bullet aperture airflow parameters based on the computational fluid dynamics method to calculate the bore efficiency in the after-effect period of different schemes.
As shown in Fig. 8, the velocity and pressure fields of the rifled gas flow without the brake are shown.When the high temperature and high-pressure gunpowder gas in the chamber is ejected from the chamber, it expands sharply outside the chamber, forming the chamber gas flow field with a complex surge system.Due to the large pressure difference inside and outside the nozzle, the high-pressure gas in the chamber expands sharply after entering the atmosphere after exiting the nozzle, and the pressure decreases and the velocity increases, forming a low-pressure supersonic free expansion zone in the area near the nozzle.The gas compresses the surrounding air during the expansion process, forming a positive excitation wave directly in front of the free expansion zone and an oblique excitation wave on the side.The high-speed low-pressure airflow in the free expansion zone increases in pressure and decreases in speed after the positive excitation wave is formed in front of the positive excitation wave and then forms a subsonic region.As the amount of gas in the launch tube becomes less and less, the pressure becomes lower and lower, the gas flow field gradually turns from expanding to shrinking until it gradually disappears.

Numerical simulation of conventional two-cavity brake
Figure 9 shows the velocity field and pressure field of the gas flow at the rifling of the conventional type brake before improvement.When the high-temperature and high-pressure gunpowder gas produced by the intense combustion in the chamber is instantaneously ejected through the chamber and enters the receding cavity in front, it expands rapidly in the receding cavity and forms a surge, and the gas is ejected from the first side aperture after passing through the surge.Due to the large pressure difference between the inside and outside of the side aperture, the high-pressure gas in the recession chamber is ejected through the side aperture and continues to expand sharply after entering the atmospheric environment, and the gas flow pressure decreases and the velocity increases, forming a flow field containing a complex surge system in the area near the side aperture.As the projectile moves forward in the brake, the high-pressure gas inside the brake is ejected from each side aperture in turn, forming a complex flow field near each side aperture, and as the gas continues to expand, the flow fields near each side aperture are coupled with each other, and the excitation systems in the flow field intersect and reflect, gradually forming a more complex flow field.In the process of gas ejection from the side aperture, the force acting on the front wall surface of the side aperture is much larger than the force acting on the rear wall surface, and the combined force of the walls of the brake in the after-effect period is forward to play the role of rejection.When the projectile flies out from the front of the brake, part of the gas in the bore is ejected from the central orifice in front of the brake, and because part of the gas has flowed out from the side orifice, the gas flow in the central orifice is greatly reduced.

Numerical simulation of improved double-cavity brake
Figure 10 shows the velocity and pressure clouds for 1 ms after the numerical simulation of the modified doublecavity brake.The principle is the same as described above.The combined force of the walls of the regenerator is forward, which acts as the regenerative efficiency.A large amount of expansion gas is ejected from the side orifices, so the final gas flow from the central bullet hole is substantially reduced.
As shown in Fig. 11, the pressure cloud diagram of the 2 ms moment comparison between the traditional and optimized brake, the pressure at the extreme point decreases by 9.8e^4 Pa, the reduction rate reaches 18%,    www.nature.com/scientificreports/ the airflow from the central bullet hole decreases significantly, the airflow pressure decreases significantly, the optimization of the double-chamber brake plays an effect.As shown in Fig. 12, the velocity cloud diagram of the 2 ms moment comparison between the conventional and the optimized brake, it is obvious that the great value of velocity decreases from 1620 to 1570 at the 2 ms moment, and the reduction rate is 3%, and the velocity of middle airflow is at the great value point, and the distribution range shrinks inward.

Comparative analysis with experiments
As shown in Fig. 13a,b, the comparison of the bore-sighted shock wave and the experimental T = 1 ms pre-bullet excitation field is shown.Since the external environment is still air and the internal airflow is only driven by a single initial jet, the wavefront surface of the shock wave is always spherical.The accuracy of the numerical simulation is demonstrated by the figure comparing the point explosion shock wavefront surface.
As shown in Fig. 13c,d for the borehole shock wave with the experimental T = 2 ms pressure farther field comparison.After the interaction between the rifled shock wave and the jet, the shock wave propagates to the farther side of the rifling.At this point, the restraining effect of the rifled shock wave on the jet is greatly weakened, and the supersonic jet acquires full development and enters a stage from relative stability to continuous decay; the projectile penetrates out of the Mach disk into the coronal air mass region, which has no stalling effect on the jet.The airflow after the rifled shock wave has stopped due to the decreasing pressure at the rifled mouth and the energy replenishment, and the rifled shock wave starts the free decay phase relying on its own energy to expand outward.
As shown in Fig. 14 for the comparison of test and numerical simulation, at this time in the numerical simulation of the surge structure is still the main axis, the surge area is also gradually expanding, with the forward  motion of the projectile, the Mach disk is gradually changed from the disc to the vertical axis, and the surge mechanism is being shaped from the main axis into an ellipsoid shape.The gas jet from the double chamber orifice rapid jet expansion, jet tail depression, in the edge of the edge of the orifice appeared in the extreme value of the surge point.As the projectile gradually breaks through the Mach disk, the subsequent shock head will be reduced, and the shock wave structure will develop to the bowl.The test of the real situation is basically consistent.As shown in Fig. 15 for the numerical simulation and test to monitor the bowl cross-section pressure curve of the brake, as shown in the figure, the optimized bowl wall pressure is significantly reduced, and the optimized type and the traditional type of brake pressure change curve are basically consistent with the test, which can prove the accuracy of the numerical simulation.

Analysis of numerical simulation results
From Fig. 16a, it can be seen that the maximum velocity of the pyrotechnic gas flow in the first side orifice section of the conventional type brake is about 1600 m/s, and it rapidly decreases to about 1000 m/s in a very short time.The maximum velocity of the optimized powder gas flow in the first side aperture cross-section is about 1100 m/s, and the decreasing trend of the optimized first side aperture velocity is obvious.
From Fig. 16b, it can be seen that the velocity of the exit section of the second side orifice of the conventional type is about 810 m/s at maximum.With the decrease of gas pressure in the receding cavity, the airflow velocity of the exit section of the side aperture gradually decreases.After optimization, the exit velocity of the second side orifice is higher than that of the conventional type.
As can be seen from Fig. 17a, after the projectile exits the chamber, the powder gas in the chamber is ejected out of the chamber cross-section and enters the first receding cavity of the brake to expand, and then ejected from the first side orifice.In this process, the maximum pressure at the front wall of the first side orifice of the conventional type is about 12 MPa, and the maximum pressure of the optimized type is about 8 MPa, and the trend of the front wall decreases obviously after optimization.From Fig. 17b the wall pressure on the back wall of the first side orifice is subjected to a larger wall pressure fluctuation and force compared to the pre-optimization, the pressure on the back wall of the side orifice is elevated.
As shown in Fig. 18a, the comparative pressure curves of the front and rear wall surfaces of the second side aperture are shown.From the curve, it can be seen that the gunpowder gas in the chamber is ejected out of the rifling section into the first receding chamber of the brake to expand, and part of it is ejected from the first side aperture and part of it enters the second receding chamber to continue to expand.In this process, the maximum pressure at the front wall of the second side orifice before optimization is about 2.0 MPa, and the maximum pressure at the front wall of the second side orifice after optimization drops to 1.7 MPa.The overall decreasing trend is smooth and similar, and it drops to a low point and stabilizes within 10 ms.From Fig. 18b, it can be seen that the maximum wall pressure before and after optimization is about 1.0 MPa.After optimization, the maximum pressure at the back wall of the brake is reduced to 0.7 MPa, and the overall decreasing trend is similar to that at the front wall, which gradually decreases to a low point and remains stable.

Experimental protocol and setup
The experimental apparatus available to the public for this shooting experiment is shown in Fig. 19.In this shooting experiment, a total of 8 rounds were fired in two states, and four rounds were fired from each of the conventional and optimized brakes.
A certain type of cannon was used in this shooting, with a maximum range of 15650 m.The maximum range was 15,650 m, with a high and low firing angle of -7° to + 35°, an average firing rate of 15-20 rounds/min, a full marching weight of 1725 kg, a 54° directional firing range, and a gun crew of 8.
A schematic diagram of this shooting experiment is shown in Fig. 20.The weather conditions were clear, with good visibility, and the gun mount was a special fixed mount.The experimental equipment was a high-speed camera, canopy target, and laser vibration measurement system.
The experimental method was to clamp the gun launcher on a special fixed stand and use a pull cord to fire.The laser vibrometer was used to simultaneously measure the recoil speed, initial velocity, and fall speed of the recoil body.High-speed photography of the airflow and body tube movement was also conducted.
Table 2 shows the initial velocity of the projectile, the recoil velocity at the exit of the projectile, and the recoil velocity at the end of the recoil period measured by the laser vibrometer for each of the four rounds of the optimized and conventional models.The difference rate was 3.56%, while the conventional type had the lowest initial velocity of 780.80 m/s and the highest velocity of 807.00 m/s.The minimum velocity of the conventional type projectile exiting the chamber was 778.90 m/s and the maximum was 795.90 m/s, with a difference rate of 2.18%.The highest recoil velocity of the optimized recoil body at the projectile exit was 6.35 m/s and the lowest was 6.12 m/s, with a difference rate of 4.4%.Compared with the conventional recoil body, the minimum recoil velocity at the projectile exit was 6.57 m/s and the maximum recoil velocity was 6.72 m/s, with a difference rate of 2.13%, which was an increase of 2.27%.The minimum recoil velocity of the optimized recoil body at the end of the after-effect period was 4.07 m/s and the maximum was 4.33 m/s, which was 1.05 m/s and 0.99 m/s lower than that of the conventional type.25.80% and 22.86% lower respectively, with a significant reduction effect.

Results and analysis
As Fig. 21 shows the comparison of the recoil velocity between the optimized and conventional firing experiments.After monitoring the recoil velocity of the gun frame assembly, barrel assembly, and puller handle, the peak recoil velocity of the first round with the optimized brakes is 0.31 m/s lower, the second round is 0.21 m/s lower, the third round is 0.91 m/s lower, the fourth round is 0.18 m/s lower, and the average is 0.40 m/s lower than that of the conventional brakes.The recoil speed extreme value difference decreased by 0.44 m/s for the first shot, 0.11 m/s for the second shot, 0.27 m/s for the third shot, 0.19 m/s for the fourth shot, and 0.25 m/s for the average.
Figure 22 shows the comparison of the recoil displacement between the optimized type and the conventional type of shooting experiment for this shooting experiment.After monitoring the recoil velocity of the gun frame assembly, barrel assembly, and puller handle, the peak recoil velocity with the optimized brakes was reduced by  From the momentum theorem, the free recoil body momentum during the internal ballistic period is obtained as: The free recoil impulse during the after-effect period is: Then the total momentum of the recoil body free recoil is:  www.nature.com/scientificreports/There are two methods of calculating the efficiency of the braking brake: the energy method and the impulse method.For small-caliber weapons, the influence of the braking brake quality factor is greater, and the impulse method is mostly used for calculation.The momentum method is used to calculate the efficiency of the brake as follows In this paper, the momentum method is used to calculate the efficiency of the brake for each scheme.Calculate the decommissioning efficiency of the conventional type decommissioner as: Calculate the decommissioning efficiency of the optimized decommissioner as: According to Table 3 above: With the same projectile mass of 0.098 kg and the same charge, the recoil momentum and the total momentum of the free recoil body in the inner ballistic period were both kept the same by the numerical simulation with the fluid calculation software.After optimization, the maximum velocity with recoiler was reduced by 0.35 m/s compared with that before optimization, and the recoil impulse with recoiler was reduced by 5.27kgm/s during the after-effects period, which was calculated to be 39% of the recoil efficiency of the optimized type by numerical simulation, which was 4% higher than that of the conventional type.As shown in Fig. 23, Compared with the measured value of 49.5%, the error value is 10.5%.The error value is 6.3% compared to the measured value.

Conclusion
In this paper, the air velocity and cross-sectional pressure of the first and second side orifices of the double-cavity brake were monitored and plotted numerically.After the shape optimization, the same numerical simulations were performed and compared with the pre-optimization one by one.After that, the recoil generator before and after optimization was subjected to a four-round firing experiment using a high-speed camera and a laser vibrometer.The recoil velocity and displacement obtained from the experiments were plotted separately for visual comparison, and the efficiency was finally calculated by the momentum method.The following conclusions were obtained: 1.The optimized side aperture exit cross-section pressure is significantly lower than the optimized exit crosssection pressure, and the reduction of wall pressure after the improvement indicates that the optimized recoil-making efficiency is effectively improved.2. The optimized type of recession maker makes the recoil impulse of the recession maker effectively reduced compared with that before the improvement, and the recession maker efficiency is significantly better than that of the conventional double-cavity recession maker.
(    www.nature.com/scientificreports/ 3. The pressure of the first side aperture of the double-chamber brake is much greater than that of the second side aperture, and the impact on the first side aperture is significantly greater during the firing process.4. The recoil velocity and displacement obtained from the experimental apparatus under good meteorological conditions, four rounds of firing experiments were conducted on each of the guns equipped with the conventional and optimized type of recoilers, and it is obvious that the optimized type of recoilers reduces the recoil displacement and the recoil velocity significantly, which effectively reduces the recoil impulse.  5.The difference between the numerical simulation efficiency and the experimental efficiency of the conventional type recoil brake is 6.3%, while the difference between the numerical simulation efficiency and the experimental efficiency of the optimized type recoil brake is 10.5%, which is generally in line with the expected assumption.6.After the gun is fitted with the double chamber brake, the maximum recoil velocity is effectively reduced, and the resulting recoil impulse and overall recoil impulse.The forward impulse generated by the recoiling body

Figure 1 .
Figure 1.Modeling of the brake.

Figure 8 .
Figure 8. Pressure field of gas flow velocity at the orifice without brake.

Figure 9 .
Figure 9. Velocity pressure cloud at different moments.

Figure 11 .
Figure 11.Comparison of T = 2 ms pressure clouds before and after optimization.

Figure 12 .
Figure 12.Comparison of T = 2 ms speed clouds before and after optimization.

Figure 13 .
Figure 13.Rifled shock wave and experimental comparison cloud.

Figure 14 .
Figure 14.Comparison of numerical simulation and test.

Figure 15 .
Figure 15.Comparison of three rifled cross-section pressures.

Figure 17 .
Figure 17.Comparison of wall pressure before optimizing one side of the orifice.

Figure 18 .
Figure 18.Comparison of wall pressure after optimizing the two-sided orifice.

Figure 19 .
Figure 19.Part of the displayable experimental apparatus.

Figure 20 .
Figure 20.Schematic diagram of the experimental principle.

Figure 21 .
Figure 21.optimization type and traditional type shooting experimental recoil body speed comparison.

Figure 22 .
Figure 22. optimization type and traditional type shooting experiment recoil body displacement comparison.

Table 2 .
Recoil body recoil velocity and projectile initial velocity test results.

Table 3 .
Comparison of efficiency calculations for brakes.